30th of the Month

Date: 06/18/2001 at 10:10:50
From: selim
Subject: Calendar tricks

Hi,

My question is, what day does the 30th of the month most often fall on 
(Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, or Sunday)?

Date: 06/18/2001 at 12:14:56
From: Doctor Peterson
Subject: Re: Calendar tricks

Hi, Selim.

This will take some work, but you can get ideas for approaching the 
problem from this similar question, about which day of the week is 
more common for Jan 1:

   Years That Start on Sunday
   http://mathforum.org/dr.math/problems/lane.10.2.00.html   

Also, check our Calendar FAQ:

   The Calendar and the Days of the Week
   http://mathforum.org/dr.math/faq/faq.calendar.html   

If you need more help, please write back and show me the work you have 
done.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 06/18/2001 at 12:48:47
From: Doctor Rob
Subject: Re: Calendar tricks

Thanks for writing to Ask Dr. Math, Selim.

The Gregorian calendar repeats with a period of 400 years. This period 
of time contains D days, where

   D = 400*(365 + 1/4 - 1/100 + 1/400),
     = 146097,


20871 weeks, and 4800 months. 4400 of these months contain a 30th.
There are 14 different yearly calendars, 7 for regular years (R) and 
7 for leap years (L) (7 because January 1 can fall on any of the 7 
days of the week). Here is a table of how many 30ths fall on each day 
of the week for each of the 14 calendars:

     Type   num     Sun   Mon   Tue   Wed   Thu   Fri   Sat
     ----   ---     ---   ---   ---   ---   ---   ---   ---
   R, Sun    43       2     2     1     1     2     1     2
   R, Mon    42       2     2     2     1     1     2     1
   R, Tue    44       1     2     2     2     1     1     2
   R, Wed    42       2     1     2     2     2     1     1
   R, Thu    44       1     2     1     2     2     2     1
   R, Fri    42       1     1     2     1     2     2     2
   R, Sat    43       2     1     1     2     1     2     2

   L, Sun    15       2     3     1     1     1     2     1
   L, Mon    14       1     2     3     1     1     1     2
   L, Tue    14       2     1     2     3     1     1     1
   L, Wed    15       1     2     1     2     3     1     1
   L, Thu    13       1     1     2     1     2     3     1
   L, Fri    16       1     1     1     2     1     2     3
   L, Sat    13       3     1     1     1     2     1     2
            ---     ---------------------------------------
            400     625   632   625   632   627   628   631 = 4400

So it seems that the 30th of the month occurs most frequently on
Monday or Wednesday (632/4400 = 14.3636% of the time), and least
frequently on Sunday or Tuesday (625/4400 = 14.2045%).

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/